stands for the spin-current density between the \(i\) -th and \(j\) -th atomic planes. Additionally, \(\vec{\sigma } = (\sigma _x, \sigma _y, \sigma _z)\)  is a vector of Pauli matrices, \(H_{ij}\)  is hopping matrix between sites \(i\)  and \(j\) , \(G_{ij}^{<}\)  is the lesser Green function of the whole coupled system and the \(\textbf {k}_{||}\)  integration is performed over the 2D in-plane Brillouin zone (BZ). The total spin transfer torque acting on the top magnetic lead is \(\textbf {T} = \sum _{i \in R} \textbf {T}_{i} = \textbf {Q}_{0,1}\) , corresponding to the spin current density between the last atomic layer of the insulating spacer (0-th layer) and the first atomic layer of the top magnetic layer (1-st layer). It is important to mention that the angular dependence of current-induced torques in MTJ devices is dictated only by the sine of relative angle, \(\theta \) , between the magnetization of the two ferromagnets [1]}, [2]}, [3]}, [4]}, [5]}, [6]}. Thus, we only need to evaluate the voltage dependence of these torques for a single angular configuration, say \(\theta = \pi /2\) , in order to obtain their dependence over all important parameters for our analysis. When the system is driven out of equilibrium under an applied voltage \(\mu _{\rm {T}} - \mu _{\rm {B}} = eV\)  we assume that the potential drops linearly inside the oxide layer.
